A Simple Counterexample to Kouchnirenko's Conjecture

Bertrand Haas

Abstract: In the late 70's A. Kouchnirenko posed the problem of bounding from above the number of positive real roots (that is, with positive coordinates) of a system of $ k $ polynomial equations in $k$ variables not in function of the degrees of the polynomials (like in Bezout Theorem), but in function of the number of terms involved, and he conjectured an upper bound. Although he never wrote this conjecture (as far as I know), many references to it can be found [K] for one of the first and [S] for one of the most recent). I recently learned that Kouchnirenko himself was "100% sure" that his conjecture was false, since a colleague of his once presented him a simple counterexample, a system of two threenomial (polynomials with three terms) equations in two variables with $5$ positive roots while the conjecture predicts at most $4$. His colleague tragically died soon afterward. The counterexample was lost and Kouchnirenko never found it again. Here such a counterexample is presented, maybe the lost counterexample found again...